# Robotics Kinematics Lesson 1

Notes for ME5421

This Blog will continue to be updated as the course progresses. Notes, code and ideas will be added as I go along. Check out the course notes at the top ↑ ;)

## Introduction

Introduction, Spatial Descriptions and Transformations, Manipulator Forward and Inverse Kinematics, Mechanics of Robot Motion, Static Forces and Torques

Lecturer: Prof Marcelo H. Ang Jr

## Contents

### 1. Introduction, Spatial Descriptions and Transformations

Robot definition. Robot classification. Robotics system components. Notations. Position definitions. Coordinate frames. Different orientation descriptions. Free vectors. Translations, rotations and relative motion. Homogeneous transformations.

### 2. Manipulator Forward and Inverse Kinematics

Link coordinate frames. Denavit-Hartenberg convention. Joint and end-effector Cartesian space. Forward kinematics transformations of position. Inverse kinematics of position. Solvability. Trigonometric equations. Closed-Form Solutions. Workspace.

### 3. Mechanics of Robot Motion

Translational and rotational velocities. Velocity Transformations. The Manipulator Jacobian. Forward and inverse kinematics of velocity. Singularities of robot motion.

### 4. Static Forces and Compliance

Transformations of static forces and moments. Joint and End-Effector force/torque transformations.

## Start of Lesson 1

Overview:

Advice: understand all the time, ask questions at real-time! understand + pratice problem solving try sample questions3 contact TA for research

Robotics need knowledge from: [ME, EE, CS] + Social Science (Understand Humanity) + Business (Commercial) + Ethics

Robots do things that

1. too intelligent to do
• dangerous
• dull
• dirty
• demeaning
1. Human cannot do as well as machine
• precision

Autonomous Driving is both!

A Typical Robot System: Sensing+Understanding, Planning, Exceuting, Learning

## Basic Kinematics

some manipulater:

• hand: end-effecter
• embrace: whole body manipulation

### Workspace

The space reachable by last link is called the workspace(sometimes, arm workspace)

Arm (primarily resposible for corase positioning) link 123(Cartesian Robot) + Wrist (fine positioning) link 456

Orientation worksapce: set of all possible orientations reachable

### Cartesian， Cylinrical， Spherical Robot

T: translation joint; R: rotation joint

Cartesian Robot: TTT RRR

Cylinrical Robot: RTT RRR

Spherical Robot: RRT RRR

So when we are talking about spherical or cylindrical or cartesian robot, we are talking about the first 3 links, i.e. arm workspace ( sometimes changing size of link will be same in kinematics, but not in dynamics )

### SCARA: Selective Compliant Assembly Robot Arm: RRT RRR; ISO 8373:2012

accerleration: m + b + k + G (mass: overcome inertia, damping: overcome friction (air, water, etc.), stiffness: if there is a spring in the system, gravity)

SCARA can be fast because the first two links do not need to overcome gravity GearBox(theta_N to theta_1): more torque, less speed, (most of the motor has gearbox, which are not back-drivable, or else it will be back-drivable) Advantage for back-drivable: motor-generator; human-robot interaction; hitting will turned into voltage change, providing compliance So SCARA is seletivly comliant, the first two links are back-drivable Back-drivable with gearbox: add a sensor (put is a the end of the pole)

### Articulated Robot Arm: RRR RRR

Rotational joint can change orientation and position at the same time while prismatic joint can only change position

Human is fully articulated (waist, shoulder, elbow, wrist, fingers)

When using a joint, one should consider: joint style and joint range motion, because some joint have limited range of motion

6 joints because 6 DOF, 6 independent variables -> 6 DOF

sphereal joint = 3 DOF! because 3 independent variables (human shoulder, wrist)

if you have more joints that 6, you can have redundant motion, like stay your hand put but moving elbow, which is great

• increase dexterity
• increase robustness
• increase flexibility

### Parallel Robot

robot that has more than one end-effector, like with two arms

### Collaborative Robot (Co-X, CoBot)

ISO: robot will stop when hit something universal robot will sense the current to stop, but not very accurate, and it is not back-drivable impedance control: if you have force torque sensors in joints, you can control the dynamics that is not physical appearant mass, appearant stiffness, appearant damping

## Chapter 1: Introduction, Spatial Descriptions and Transformations

Position of point O in frame A $$^AP_O = \begin{bmatrix} ^AP_{OX} \\ ^AP_{OY} \\ ^AP_{OZ} \end{bmatrix} \in \mathbb{R}^3$$

#### Rotataion Matrix

If object B is in frame A and the frame B attached to B is $^Ai_B, ^Aj_B, ^Ak_B$ , then $$^AR_B = \begin{bmatrix} ^Ai_{Bx} & ^Aj_{Bx} & ^Ak_{Bx} \\ ^Ai_{By} & ^Aj_{By} & ^Ak_{By} \\ ^Ai_{Bz} & ^Aj_{Bz} & ^Ak_{Bz} \end{bmatrix} \in \mathbb{R}^{3 \times 3}$$ $$det(^AR_B) = +1$$ R is a Proper Orthogonal Matrix $$^AR_B^{-1} = {^AR_B^T} = {^BR_A} \\ {^AR_B} {^AR_B^T} = {^BR_A} {^AR_B} = I$$ But only three parameters are needed to describe the rotation matrix, so we can use Euler Angles, Axis-Angle, Quaternions, etc.